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TecHnical RepoRT no. 1

Measurement and Computation of the new Oeresund LoopBy Casper Jepsen and Klaus Schmidt

DANISH MINISTRYOF THE ENVIRONMENT

National Survey and Cadastre

Measurement and Computation of the new Oeresund Loop

Casper Jepsen and Klaus Schmidt

National Survey and Cadastre – Denmark 8 Rentemestervej

DK-2400 Copenhagen NV Denmark

caj@kms.dk kes@kms.dk http://www.kms.dk

s

Casper Jepsen and Klaus Schmidt:

Measurement and Computation of the new Oeresund Loop

National Survey and Cadastre, technical report series no. 1 ISBN 87-92107-09-5Technical Report Published 2008-06This report is available from www.kms.dk

TECHNICAL REPORT NO. 1 / 2008

Contents

1. Introduction 11.1 Historical background 11.2 Purpose and planning of the new measurements 11.3 The construction of the Link 2

2. The measurements 32.1 On-shore levelings 32.2 The Link 32.3 The optical water crossing (1980) 4

3. Lining up the Oeresund Loop 53.1 The points for temporal connection 53.2 Debugging the Link 53.3 Configuration 5

4. Analysis and adjustment 64.1 Double run discrepancies 64.2 Adjustment 6

5. Loop closing errors and land uplift 7

6. The loop set up from optical and hydrostatic water crossing 8

7. Gravity processing 87.1Preliminary remarks 87.2 Particular circumstances 97.3 Bouguer anomalies applied 97.4 Gravity interpolation 97.5 Orthometric correction 9

8. Conclusions 10

Appendix1 11Leveling and gravity1. Geopotential numbers and orthometric heights. 112. Conversion of leveled height differences. 113. Gravity reduction by means of Bouguer plates. 124. Bouguer gravity and anomalies. 135. Gravity interpolation (GeoGrid) 136. The computation of mean gravity 147. Dataflow 15

Appendix 2 161. Stability investigation in the Helsingør region 162. Gravity along the Link. 17

References 18

Double runs of the Oeresund Loop 19

Technical report handed over to the NKG Working Group for Height determination at the meeting in Akranes, Island, June 13-14, 2005

Abstract. In order to connect the height systems of Denmark and Sweden the Sound has been crossed repeatedly in the past at its northernmost end. From this, two small local loops could be set up from optical and hydrostatic water crossings. However, due to the recent completion of the Oeresund Link a new crossing could be established at the southernmost end of the Sound, giving for the first time the possibility to close a loop around the Sound. The crossing has been accomplished by motorized leveling in the year 2000 as a joint enterprise by National Land Survey of Sweden (NLS) and National Survey and Cadastre of Denmark (KMS).

The setup of the new loop is carefully defined and a description of all measurements involved is given. The measuring quality of the new crossing is shortly analyzed and compared to the Danish Third Precise Leveling. Based on the closing errors of the new loop and a local loop, which can be set up from an optical crossing and a hydrostatic leveling as well, conclusions are drawn on the reliability of the crossings involved and their usefulness for connection of the national height systems.

For the sake of consistency gravity has been applied to the leveling, however, for practical purposes the effect is almost insignificant. This also goes for postglacial land uplift corrections.

Acknowledgement: Thanks to our Swedish colleagues from the NLS for providing the Swedish leveling data, in particular to Per-Ola Erikkson always available to answer new questions. Thanks also to J. Mäkinen from the Finnish Geodetic Institute (FGI), delivering the uplift grid, K. Johansen (KMS), setting up the on-shore leveling line from the Danish Third Precise Leveling, and G. Strykowski (KMS) for discussing gravity reduction using the GeoGrid program.

1

1. Introduction1.1 Historical background. In order to connect the height networks of Sweden and Denmark, i.e. the

connection of the continental European networks with the networks of the Nordic countries, the Sound has to be crossed. Nowadays, this possibly could be achieved with sufficient accuracy by GPS, formerly, however, other solutions had to be found. Thus, the Sound was crossed by two optical water crossings and a hydrostatic leveling all carried out at the narrowest place (5km) at the northernmost end, Helsingør/ Hälsingborg. (For the sake of completeness three additional optical crossings from 1896/1898 and 1981 should be mentioned, se the references below. However, all of them are of doubtful quality.)

The first optical water crossing connecting the national First Precise Leveling networks took place as early as in 1896/1898, cf. the crossing K-T in Den Danske Gradmaaling (1909). As such crossings can be highly affected by deflections of the vertical as well as asymmetrical refraction the national Second Precise Leveling networks were connected by a hydrostatic leveling in 1939, cf. Nørlund (1946). Indeed, comparing these crossings a major deviation of 17mm turned out, ibid. p.81. Of course, this could be caused by poor quality of the optical crossing, but on the other hand the Sorgenfrei/Tornquist fault zone, separating the Fennoscandian Shield from the Norwegian Danish Basin, is crossing the Sound just in the area considered, cf. figure 1. Thus, irregular vertical movements of the reference benchmarks, occurred in the intervening time span of about 40 years, could not be excluded. Besides, due to the postglacial Fennoscandian land uplift, cf. Ekman (1996), there are no stable points in the Nordic countries. Nevertheless, if this is the only cause of a point’s vertical movement we shall use the term “stable” throughout this report.

In order to investigate this further a new optical crossing, recorded in Bedsted Andersen et al. (1986), was established in 1980, by the Danish Geodetic Institute (G.I.) and the Swedish NLS. As decided by the Leveling Working Group of the Nordic Geodetic Commission (NKG) the new crossing was designed as a repetition of the old one performed under the same conditions, as far as possible. Supposing this would imply the same systematic errors the difference of both crossings, based on a time span of about 100 years, could give an indication of the stability across the Sound. This concept possibly has been successful, because recomputing the old crossing and comparing it with the new one from 1980 a deviation of just 2mm was found, cf. Bedsted Andersen et al. (1986), p.35. As this result is subject to an estimated mean error of 5mm it was concluded “that no detectable relative height movements have taken place across the Sound from 1896 up to now (1980)”. However, the deviation is based on the somewhat doubtful leveling from 1984 in table 6; using the value from 1980/81 instead the difference would be 7mm.

figure 1: Location of the Sorgenfrei-Tornquist zone in the area of the Sound, from Abramovitz, T. et al. (1997)

1.2 Purpose and planning of the new measurements. In the year 2000 a new possibility to cross the Sound came into existence due to the completion of the Oeresund Link (16km) connecting the cities of Copenhagen and Malmö, cf. sect. 1.3 for constructional details. Based on a proposal of the Height Determination Group of the NKG in summer 1999, a common decision was made by NLS and KMS to use the Link for the connection of the new national Third Precise Levelings conducted during the last two decades. If the height determination along the Link could be done with sufficient accuracy the height connection between Sweden and Denmark would be strengthened. Moreover, connecting the Link to the new leveling lines along the Sound and using the optical crossing from 1980 it would be possible for the first time to close a loop around the Sound, cf. figure 2, in order to check measuring quality. Hence, the primary goal of this report is the computation of the closing error taking into account postglacial uplift of the benchmarks

2

occurred within the measuring time span of the loop. Moreover, we shall compute the closing error of the local loop at Helsingør/Hälsingborg, which can be set up from the optical crossing (1980) and the hydrostatic leveling.

figure 2: The Oeresund Loop and its points for temporal connection

As the networks to be connected are coming from motorized geometric leveling it was clear from the beginning that the height determination of the Link should be conducted in the same way. For control purposes the measurements on the high bridge should be supplemented by motorized trigonometric leveling with sight lengths up to 500m to avoid all the intermediate setups needed for geometric leveling. However, due to the open sea strong wind is frequently occurring in the area. Also, heavy machines for road construction would be working during the measurements. Thus, it was doubtful if the leveling could be done with sufficient accuracy within the short period of time granted by the Oeresundsbro Konsortiet. Nevertheless, the completion of the Link was a one-time opportunity to strengthen the height connection across the Sound, therefore the measurements were started in the beginning of April, 2000.

1.3 The construction of the Link. According to information from the Internet, www.oeresundsbron.com, the Link is consisting of a 4 km long tunnel, an artificial island, Peberholm (4 km), and an 8 km long bridge made up by the western approach bridge (3 km), the cable-stayed high bridge (1km) with a free span of 490 m, and the eastern approach bridge (4 km), cf. figure 3. The tunnel is necessary for security reasons due to the airport of Kastrup nearby.

figure 3: High Bridge and approach bridges of the Oeresund Link

3

Inside the tunnel (width 40m, height 7m, lowest point -20m) a 4-track motorway and a double track railway are running side by side through 4 tubes immersed in a ditch at the bottom of the sea and covered by stones. The soil from the immersion of the tunnel has been reused to build up the island of Peberholm, surrounded by shallow water and measuring about 300m across. The island is needed for the transition of the four tunnel tracks to the two levels of the bridges with the motorway on the upper deck about 10m above the railroad level. Motorway and railroad deck are connected through a stiff steel framework. The approach bridges are supported by piers, whereas the high bridge (max. height 70m) is tied to two pair of pylon legs, 204m high.

2. The measurements2.1 On-shore levelings. Except for a short line of precise geometric foot leveling, connecting the optical

crossing from 1980 and the Danish leveling network of that time at Helsingør, the on-shore lines of the new loop are part of the national Third Precise Leveling networks. Both networks have been measured by geometric motorized leveling, cf. Becker (1987), i.e. rods and level are moved by vehicles specially designed for transportation, and levels of the type Zeiss NI 002 as well as calibrated invar rods are applied in accordance with the Nordic guidelines for leveling, cf. Becker and Bedsted Andersen (1986). As regards precision motorized leveling is comparable to foot leveling, i.e. a mean error of about 1mm of 1km double run as found from the adjustment of the Danish Third Precise Leveling, cf. Schmidt (2000). However, productivity, essential for the measurement of the Link, is much higher, say, 2 km single run per hour.

In precise leveling the proper observation of a section height difference is a double run, i.e. the mean of one measurement (single run) in either direction. On location the discrepancy ρ between forward and backward run is tested. If the absolute value of ρ’ = ρ / L is exceeding a certain limit k both runs are rejected and a new double run has to be measured. Here ρ is in mm, L is the section length in km, and k=1.8or k=2.0 for Danish or Swedish leveling.

2.2 The Link. The leveling of the Link has been accomplished by one team from either country, using the same procedures and equipment as for the national Third Precise Levelings. The work took place within 8 days in chilly weather (3-8 C˚). Even though the leveling was planned from day to day in accordance with the weather conditions some of the measurements had to be conducted under difficult circumstances, e.g. strong wind in plain sunshine.

Wherever permitted by the Oeresundsbro Konsortiet proper leveling bolts were established by the national road authorities. Otherwise, removable bolts in the tunnel or tops of solid vertical screws were used for leveling control. Along the approach bridges the control points were pointed out on every second pier corresponding to leveling sections of 480 m length. Stable leveling points kept in the database of KMS are indicated by bold types in table A4. Of course, the piers were used for support of the level and the rods during the leveling. As a rule, neighboring leveling points were connected by 5 or 6 double runs.

Trigonometric leveling of the high bridge was carried out by an extra Danish team. The results were in good agreement with the geometric leveling, thus it was concluded that the latter was influenced by vertical movements much less than expected. For different reasons the trigonometric results have not been used in our computations.

Finally, as recorded by R. Forsberg (KMS) gravity was measured at 6 new points, cf. app.2, sect.2.1, in order to improve gravity interpolation along the Link, which is necessary for a consistent leveling adjustment.

figure 3.1: Motorized leveling on the High Bridge

4

2.3 The optical water crossing (1980). The crossing performed during 2 weeks in the autumn of 1980 is connecting the top of the Telegraph Tower of the Castle of Kronborg with the top of the cliff of Pålsjø, a few kilometers north from Hälsingborg, cf. figure 4. These sites have been used, too, for the first optical crossing. They are located at a maximum height (30 m) above sea level in order to lessen asymmetrical refraction at the observation sites.

figure 4: Special locations in the area of Helsingør/Helsingborg

The crossing has been accomplished applying the so-called fjord crossing method often used in Norway,i.e. the height angles of a pair of special target plates, attached to a leveling rod raised close to the opposite site, are measured simultaneously from both sites by means of the scale of the tilting screw of a Wild N3 level, cf. figure 5. From this, the height difference of the sites’ reference benchmarks can be computed twice. As can be shown each of these values, which are affected by the earth’s curvature, the level’s collimation error, and refraction, can be interpreted as an ordinary leveled setup height differences (backward - forward reading) with leveling rods fictitiously raised at the reference benchmarks, however, the sight lines are of very different lengths. Further details on measurements and computations can be found in Bedsted Andersen et al. (1986).

figure 5: Optical water crossing from the top of Telegraph Tower aiming to the signal plates in Sweden

5

It turned out from the computations that the height angles were highly affected by unsymmetrical refraction. Consequently, all readings taken under unfavorable meteorological conditions, i.e. about half of the entire number, were excluded from the final computation.

Last but not least, the benchmark on top of the Telegraph Tower had to be connected to the existing leveling network by measuring its height above the benchmark at the bottom of the tower. This was achieved through a vertical steel tape connecting both points. The length of the tape corresponding to a given traction and temperature was found from interferometer measurements.

3. Lining up the Oeresund Loop3.1 The points for temporal connection. In order to set up the loop leveling lines from different decades

had to be connected. Wherever these levelings are overlapping any point of the overlap could be chosen for connection, however, the vertical movement of the connection point should be estimable as good as possible, cf. sect.5. With this in mind the connection points, shown in fig.2, have been carefully selected.

On the Danish side points preferably joining a group of subsoil benchmarks should be used for connection purposes. Frequently, the benchmarks of such group are standing in a row within a short distance, say, 50m. They always are leveled jointly to monitor vertical stability. In general, a subsoil benchmark is consisting of a well-founded granite pillar or a concrete block, the top of which is up to 1½m beneath the ground. Its proper motion (relative to the surrounding soil) is expected to be insignificant. Also, movements, which are nonlinear in time, are unlikely. Thus, subsoil benchmarks are considered as the most stable kind of leveling marker in Denmark.

Due to the lack of measurements there is no subsoil point in the area of Copenhagen to connect the Link with the Danish Third Precise Leveling. Anyhow, there are three different bolts, which could be used. Among them we have selected the bolt K -01-06817 located at the old, massive building of the Castle of Christiansborg (The Danish Houses of Parliament). According to preliminary results from the computation of land uplift in Denmark based on the three Danish Precise Levelings, the bolt seems stable relative to the nearest subsoil benchmark G.M.1393. However, both points could be on unstable ground, because they are close to the sea in a few meters height. Nevertheless, the bolt also seems stable relative to the remote subsoil group G.I.1803-1806 located inland close to the former Geodetic Institute’s laboratory.

In the Helsingør region the connection of the Third Precise Leveling and the optical crossing (1980) is complicated by inconsistent leveling results as recognized already in Bedsted Andersen et al. (1986), sect.II,1.2.2. Investigating the levelings from 1940 on, cf. app.2, sect.1, the connection point finally chosen is the benchmark G.I.1607 of the subsoil group Hamlets Grav, se fig. 4.

In addition, several lines from the Swedish Third precise leveling had to be joint in order to connect the Helsingborg area with the Link. The points of connection, 032*1*3124, 022*1*6515, and 022*1*2423, have been pointed out by our Swedish colleagues claiming stability of these points in the last few decades.

3.2 Debugging the Link. It was obvious from the data delivered by NLS that the rejection limit of precise leveling, cf. sect. 2.1, had not been applied to the Swedish double runs of the Link. Catching up on this we noticed a rather large number of severe rejections on the eastern approach bridge. It could be shown from a closer investigation that this was caused by 2 leveling points, 4078/500 501 and 4078/500 502, both alternately moving within a few millimeters between 2 vertical positions depending on the day of observation. According to the leveling teams, this probably can be explained through heavy machines for road construction working close to these points on some of the days. In our computations we have used additional point numbers, i.e. the original ones extended by .1, to distinguish between both positions. However, due to some rejections the point 4078/500 502.1 became the end point of an antenna, therefore it will not occur in this documentation. It should be mentioned that point numbers of the type 4078/…are referring to the numbering system adopted by the national road authorities,

3.3 Configuration. Apart from the optical crossing the loop has been set up entirely by double runs obtained from single runs in the national data files, the latter corrected for scale graduation errors and temperature. All the double runs included have been accepted according to sect. 2.1 except for a few runs outside of the Link. Repeated double runs have been included separately, antennas have been removed. Along the Link the loop is consisting of line-shaped networks connecting the different sections of the Link, such that neighboring networks have just 1 point in common. Outside of the Link the loop is consisting of consecutive double runs without crossovers and just a few repetitions.

In total, the loop has been set up from 315 points connected by 502 double runs, in addition to the optical crossing from 1980 and the corresponding steel tape measurement. Double runs are listed at the end of this report. The circumference of the loop is about 192 km, but due to the large number of repeated double runs along the Link the accumulated length of all double runs involved is about 254 km (76 km along the Link).

6

A schematic outline in clockwise order is given in table 1. Bold point numbers are indicating the connection points in fig. 2. In case of leveling ΣL is denoting the accumulated length in km of the double runs included. The ΔĤ-values below are adjusted Helmert height differences according to sect. 4.2, except for the water crossing, which is orthometrically corrected, whereas the corresponding steel tape measurement is uncorrected. The sources are referring to the Swedish Third Precise Leveling data, the KMS leveling data files #DK_niDniv and #DK_niDprs, the G.I. field book GA.I, vol. 102, and Bedsted Andersen et al. (1986).

location point no. ΔĤ method ΣL year data source

Hamlets Grav G.I.1607-13.65976m geom..foot lev. 2.8 1980 #DK_niDniv & GA.I, bind 102,

p.128916, 128918

GA.I, bind 102, p.129142

Bedsted A., p.30

Sw. 3rd prec. lev.

do.

do.

Link, end 4078/100 501

Christiansborg K –01-06817

do.

#DK_niDniv & Sw. 3rd prec.lev.

5.52243 do. 14.5 1998/2000 #DK_niDniv

19.31278 do. 60.5 1992 #DK_niDprsHamlets Grav G.I.1607 ------------ -------

-0.01350 258.5 km

table 1: Set up of the Oeresund Loop

4. Analysis and adjustment4.1 Double run discrepancies. In order to compare the leveling of the Link with the Danish Third Precise

Leveling we have computed sectionwise the average (av) and standard deviation (std) of the normalized discrepancies ρ’, cf. sect. 2.1, from all the double runs of the loop included in the Link. As mentioned before all these runs have been accepted. The results are shown in table 2, where the last row is indicating the percentages of rejected double runs based on the total number of runs performed along the Link (accepted or not).

location: east bridge high bridge west bridge Peberholm tunnelav(ρ’) : 0.25 mm 0.28 0.49 -0.11 -0.11std(ρ’) : 0.77 mm 0.87 0.76 0.99 0.97number : 59 15 59 32 64

reject. pct.: 13 21 9 14 2

table 2: Statistics of double run discrepancies

According to Schmidt (2000), sect.3223, the rejection percentage of the Danish Third Precise Leveling is below 5%. Hence, the values above are larger than usual, however, the indicated averages and standard deviations of ρ’ are rather common. In particular, the standard deviations along the bridges are almost identical with the average value 0.75 mm from the Danish Third Precise Leveling, cf. sect. 5322, ibid. We thus conclude that the leveling of the Link after rejection is comparable with the Danish Third Precise Leveling as regards discrepancies from forward and backward run.

4.2 Adjustment. Based on the assumption that all double runs are stochastically independent the loop can be adjusted in a simple way. Regarding the Link least squares adjustment is applied separately to the different networks connecting the sections of the Link, cf. sect. 3.3. That means, keeping fixed à priori the height of a single point of the network, ΔH12=H2-H1 is the observation equation of the double run from point P 1 to P2 of length L, and VarΔH12=σ2L is the corresponding variance. Outside of the Link multiple double runs have been adjusted by their mean. Note, prior to the adjustments double runs have been converted into Helmert height differences by adding the orthometric correction, cf. app.1, sect. 2.1 and 2.3.

The results from the sectionwise adjustments are shown in table 3, where ΔĤ is the adjusted Helmert height difference, m.e.(ΔĤ) is the corresponding estimated mean error, s is denoting the estimated mean error of 1km double run, whereas n and m is the number of double runs and leveling points involved. The last row is indicating the adjusted Helmert height difference of the Link and the corresponding estimated mean error, that is the square root of the sum of the m.e.(ΔĤ)-values squared. Note, the surprisingly low value, 1½mm, is caused by the large number of repeated double runs, cf. sect.2.2.

Telegr. tower, foot K –06-0900821.7468 steel tape 0.022 do.

Telegr. tower, top K –06-09170-0.59106 opt. water cr. 4.8 do.

Pålsjø Cliff 032*2*31162.23258 geom..mot. lev. 3.1 1980/81

Kärn., found.wall 032*1*3124-15.44517 do. 58.4 2001/02

022*1*6515-1.33355 do. 32.2 1983

022*1*24233.48312 do. 6.4 2000

Link, start 4078/500 503-21.28167 do. 75.8 do.

192 16

7

location point no. ΔĤ m.e.(ΔĤ) s n m

4078/500 503east bridge 47.59273m 0.77mm 0.77mm 59 19

4078/430 261high bridge 0.05892 0.63 1.21 15 4

4078/410 261west bridge -51.50084 0.72 0.87 59 14

4078/300 511Peberholm -18.68245 0.63 0.72 32 8

4078/300 501tunnel 1.24997 0.48 0.51 64 14

4078/100 501 ------------ ---------21.28167 1.46

table 3: Results from the sectionwise adjustments of the Link

According to Schmidt (2000), p.31, s-values in the range from 0.6 to 1.0 are expectable, thus the values above are quite common. As regards the rather large s-value of the high bridge point movements during the leveling could not be detected from the data, more likely, the value is caused by a single doubtful double run, which, however, had to be accepted according to sect.2.1.

The loop closing error has been computed by summing up the height differences in table 1. As is seen a value of -13.50 mm has been obtained. Also this is in good agreement with the Danish Third Precise Leveling. The mean error of the computed closing error can be sufficiently estimated from the dominating contribution from the on-shore leveling lines, the length of which is the length of the loop (192km) minus the length of the Link (16km). Assuming a mean error of 1 mm per 1 km double run the estimated mean error ofthe closing error is ≈ 13 mm.

Finally, with reference to possible future remeasurements of the Link, we give the adjusted height differences Δĥ between the stable leveling points recorded in the files of KMS. The Δĥ-values of table 4 are coming from ordinary adjustments of measured double runs, i.e. without applying orthometric corrections.

point no. 4078/: 500 503 400 506 400 502 300 511 300 502 300 501 100 502 100 501Δĥ : 49.40225m 0.02898 -53.28031 -16.26851 -2.41392 -1.55703 2.80697lev. distance : 4.3km 0.5 4.0 3.1 0.1 4.2 0.1

table 4: Adjusted height differences of selected points of the Link

Still ignoring these corrections a loop closing error of -12.94 mm has been found, in addition to the adjusted height difference of the Link, Δĥ = -21.28157 m. Comparing this with the results above it is obvious that the effect of the orthometric correction is insignificant from a practical point of view.

5. Loop closing errors and land uplift

The closing error in sect.4.2 has to be corrected for vertical movements of the connection points. In order to derive a correction formula consider the loop in the figure to the left composed of leveling lines from different years ti

connecting the points Pi, and let ΔH’i,i+1(ti)=ΔHi,i+1(ti)+dε denote the corresponding orthometric height differences, where ΔHi,i+1(ti) is the true value and dε the error induced from leveling. Assuming linearity of the movements given by annual uplift rates ai, it holds

ΔHi,i+1(ti) = ΔHi,i+1(to) + (ai+1 - ai)(ti - to )where to is a certain reference year adopted à priori. Summing up along the loop the sum ΣΔHi,i+1(to) is vanishing, since orthometric heights are consistent, cf. app.1, sect. 2.1. Hence, let ao be any constant value, then it can be easily shown that the closing error corrected for vertical movements can be written

Σdε = ΣΔH’i,i+1(ti) -Σ(ai - ao)(ti-1 - ti)Note the equation is not depending on the reference year to. Obviously, the larger the time span (ti-1, ti) the better an estimate of uplift is needed.

Annual uplift rates relative to mean sea level can be computed at specific points from repeated leveling and sea level observations. However, final values in the Oeresund region computed by the national surveying authorities are not available, yet. Nevertheless, established by Ekman (1996) there exists a continuous model of Fennoscandian postglacial uplift relative to mean sea level, which we have used in our computations. Other models possibly more refined are available, but no matter which model is chosen it cannot provide the actual uplift of specific points, if they are affected by local movements or proper motions. This is the reason why the connection points of the Oeresund Loop have been selected so carefully.

The uplift values applied have been interpolated from a numeric grid based on Ekman’s map and handed over to the Height Determination Group by J. Mäkinen (FGI). They are corresponding to (a i-ao) in the formula above, where ao is the uplift of the mean sea level. The values in mm/y are given in the table below.

8

Hamlets Grav Kärnan, found. wall Kärnan, tower chamber Christiansborg G.I.1607 032*1*3124 032*2*3109 022*1*6515 022*1*2423 K -01-06817-0.13 -0.11 -0.11 -0.23 -0.34 -0.33

table 5: Annual uplift rates from Ekman

Using the measuring years from fig.2 we get from table 5, Σ(ai-ao)(ti-1-ti)=-0.25mm, i.e. the closing error in sect.4.2 is practically unchanged. Hence, we conclude the closing error corrected for postglacial land uplift is about -13mm, but recall, proper motions e.g. are not taken into account.

6. The loop set up from optical and hydrostatic water crossingTo line up the loop the crossings have to be connected by local leveling lines, among others the line from

Pålsjø Cliff to Kärnan, cf. fig. 4, which has been measured repeatedly in the past. The results are given below. Note, the value from 1984 seems unreliable and is not used.

1896/97 1939 1956 1980/81 1984

9.217 m 9.21735 9.2176 9.21976 9.2247source: Bedsted A., p.21 Nørlund, p.25 NLS Sw.. 3rd prec. lev. Bedsted A., p.21

table 6: Height differences from Pålsjø Cliff to Kärnan, tower chamber

An outline of the loop is given in table 7, where the connection points are indicated by bold types. The Δĥ- values are adjusted measured height differences without applying orthometric corrections. The circumference of the loop is about 17 km and a loop closing error of -2.13 mm has been found from summing up the height differences. From table 5 we get, Σ(ai-ao)(ti-1-ti)=0.80 mm, i.e. the closing error corrected for postglacial uplift is about -3mm.

location point no. Δĥ method year data source

Hamlets Grav G.I.1607-13.65975m geom..foot lev. 1980 cf. table 1

Telegr. tower, foot K –06-0900821.7468 steel tape do. do.

Telegr. tower, top K –06-09170-0.5903 opt. water cr. do. do.

Pålsjø Cliff 032*2*31169.21976 geom..mot. lev. 1980/81 table 6

Kärn., tower 032*2*3109-35.81085 geom. foot lev. 1939 Nørlund, p.24

0-ref. point (S)-0.01137 hydrostatic lev. 1939 Nørlund, p.79

0-ref. point (DK)1.05335 geom. foot lev. 1939 Nørlund, p.22

City Hall K –06-0901118.94986 do. 1940 Nørlund, p.23

Hamlets Grav G.I.1606-0.89963 do. 1940 G.I. comp.vol., p.2077 ff.

Hamlets Grav G.I.1607 ------------0.00213

table 7: Set up of the loop from optical and hydrostatical crossing

7. Gravity processing.7.1 Preliminary remarks. According to theory leveling should be computed from geopotential or

orthometric height differences (both are based on gravity), but not from the field measurements, cf. app1, sect. 2.1-2.3. However, it turned out that this did not really matter, cf. fig. 8.

As we know from the Danish Third Precise Leveling the deviations of leveled height differences from corresponding orthometric height differences are hardly summing up along a loop in a low and flat area with smooth gravity like the Oeresund region. Moreover, they normally are rather small, say, less than 0.1mm per km. However, gravity behaves irregular in the area of Helsingør/Hälsingborg, cf. fig.6. In addition, the leveling of the Link certainly is incomparable with normal leveling, since measurements have been performed not only on the ground but also on the bridges and in the tunnel.

Due to these considerations we have decided to apply the strict approach, i.e. leveling combined with gravity, even though this makes the computations a good deal more cumbersome. In the present case we have used orthometric height differences, because the orthometric corrections immediately are illustrating the effect of gravity on leveling.

9

7.2 Particular circumstances. Usual textbooks on leveling are referring to leveling on the ground, thus the common computation formulas of Bouguer anomalies, gravity interpolation, and mean gravity along the plumb line had to be modified for the measurements on the bridges and in the tunnel of the Link. Computed from form. (9.3) in Blakely (1996) gravity contributions from the tunnel or the pillars/pylons of the bridges are less than 1 mgal, therefore they have been ignored in all our computations. Consequently, points on the bridges have been considered as points in free air above the sea, i.e. case [c], cf. app.1, sect.4 - 6, on Peberholm as case [d], ho=5 m, and in the tunnel as case [e], t1>0,except for the endpoints, case [e], t1<0. In addition, the point on top of the Telegraph Tower, as case [d], ho=21.75 m.

Furthermore, ground density ρ=2.00 g/cm3 has been used in all our gravity computations. As is seen from fig.6 ground conditions along the Oeresund Loop are more or less similar on both sides of the Sound, except for the Hälsingborg area, thus the traditional Danish density value above has been used in the entire region. Anyhow, the more common value ρ=2.67 could have been used, too, this does not really matter.

7.3 Bouguer anomalies applied. Fig. 6 is showing the Bouguer anomalies (ρ=2.67) available in the Oeresund region extracted from the NKG gravity database for gravity interpolation. Points from marine gravity as well as the new gravity points along the Link are not shown. As is seen the anomalies between Helsingør and Hälingborg are increasing about 24 mgal, which is indicating an uplift of the bedrock underneath Hälsingborg. In general, the anomalies on the Swedish side are growing in north-east direction, which is in good agreement with the north-west direction of the bedrock ridges Kullen (altitude 188 m) and

188 m

-20 -10 0 15 30 mgal

212 m

Söderåsen (212 m). Fig. 6: Bouguer anomalies in the Oeresund regionFor the sake of consistency the anomalies in fig. 6 have been transformed into values corresponding to

ρ=2.00, taking into account that the data points are either on the ground above the geoid or at the surface of the sea (from marine gravity), which is indicated by positive or negative heights in the database. The latter is referring to the (negative) depth of the sea. The transformation formula is given in app.1, form. (12). In addition, we have computed the Bouguer anomalies (ρ=2.00) of the new gravity points along the Link. Measured gravity and results are given in app.2, table A3.

7.4 Gravity interpolation. Applying the strict approach a gravity value is required at every leveling point. Based on the transformed Bouguer anomalies (ρ=2.00) mentioned in sect.7.3 gravity values have been found applying the GeoGrid program, cf. app.1, sect. 5.1, running the job ‘nivtyng.bat’. Since ground density ρ=2.67 is used by the program the interpolated gravity values from GeoGrid had to modified (mod.1) according to ρ=2.00 using the formulas in app.1, sect.5.2. Furthermore, depending on the input height h any interpolation point is considered by the program either on the ground above the geoid (h>0) or at the surface of the sea (h<0, -h is the depth of the sea). That means applying the formulas, ibid., further modifications (mod.2) had to be done as regards the points of the Link and the point on top of the Telegraph Tower. Concerning the Link the final results gP* are given in app.2, table A4.

7.5 Orthometric correction. Mean gravities g P have been computed from the formulas in app.1, sect.6,using leveled heights. Hereafter, the orthometric corrections have been found from form. (6), app.1, using the average gravity value go=981500 mgal. The correction has been applied, too, to the optical water crossing, but not to the steel tape measurement.

In fig.8, bold curve, we have shown the differences between adjusted Helmert heights and the heights from an ordinary leveling adjustment starting at Hamlets Grav, where we have adopted the same initial height value. As is seen the impact of gravity on the adjusted heights is in the range of about ±1 mm. Also note the short segment (6km) north of Landskrona (Lk), where the differences are increased by almost 1½ mm. The jump at the beginning of the curve is coming from the orthometric correction (-0.76 mm) of the optical water crossing (w.cr. in fig.7). It is doubtful, if the computed value is strictly correct, since form. (6), app.1, is based on leveling with forward and backward sights of equal length, however, the sight lengths of the water crossing are extremely different, say, 50 m and 5 km. Disregarding the crossing and the segment mentioned above the curve behaves as expected, i.e. smoothly increasing from north to south and smoothly decreasing in the opposite direction, cf. app.1, sect.2.3. Nothing extraordinary is happening along the Link. Note also, the final value of the curve (-½mm) is indicating the deviation of the loop closing error applying gravity or not, cf. sect. 4.2.

tun

w

1

80m604020

0-20

20

10

0

-10

-20

figure 7: Height profile of the Oeresund Loop

DK Sweden Denmark

G.I.1607 G.I.1607

0 50 100 150 200 km

diff. (Helmert - leveled heights)

Bouguer anomalies

Hb Lk

0 50 100 150 200figure 8: Differences (Helmert heights – leveled heights) in 0.1mm vs. Bouguer anomalies in mgal

Additionally, we have shown in fig.8 the Bouguer anomalies (ρ=2.00) along the loop. Reflecting the depth of bedrock below the ground the anomalies are strongly increasing from Helsingør to the opposite side of the Sound, whereas a corresponding decrease is found along the line segment mentioned above.

8. Conclusions• Applying the usual rejection limit for leveling to the measurement of the Link, it is fully comparable to the Danish Third Precise Leveling. Thus, it should be used for the connection of the new height networks in Denmark and Sweden.• The suspicion that the optical water crossing (1980) is defective cannot be confirmed by the closing errors corrected for postglacial uplift, neither from the Oeresund Loop (-13mm), nor from the loop including the optical and hydrostatic crossing (-3mm).

1

Appendix 1: Leveling and Gravity

Normally, textbooks on leveling are only referring to leveling on the ground. In the following we try to substantiate the validity of the fundamental equations, cf. form. (3), (5), and (6) below, also for leveling on bridges or in tunnels. In this case, however, the common formulas for the computation of Bouguer anomalies, interpolated gravity, etc. have to be modified as shown.

1. Geopotential numbers and orthometric heights.To move a particle of mass in a three-dimensional force field from one point to another the work required

generally is depending on the path taken by the particle. However, as can be shown the earth’s gravityacceleration g can be written as the gradient Moritz and Heiskanen (1967), sect. 2-1

(W / x, W / y, W / z) of a real function W in space, cf.

g = gradW (1)This is the fundamental equation for all what follows.

From (1) can be concluded due to the findings of Potential Theory that the work done by the gravity field is independent of the path. The function W is called the gravity potential of the earth, its value at the point P is the work done by the gravity field to move the unit mass from infinity to P, cf. Torge (1991), sect.2.1.1.

Due to (1) the work W2-W1 done by the gravity field to move the unit mass from point P1 to P2 is given by integrating the magnitude g of the gravity vector g along an arbitrary path connecting the points

P1

W2-W1 = gdnP2

(2)

Here dn is the elemental vertical displacement of differentially separated equipotential surfaces, W=constant, along the path, cf. Moritz and Heiskanen (1967), form. (4-4). The increment dn is counted positive in outwards direction, i.e. opposite to the gravity vector.

Let the geoid be given by the equipotential surface W=Wo then the geopotential number CP of any point P in space is defined by

CP=Wo-WP

Hence, CP can be interpreted as the work done by the gravity field to move the unit mass from the point P to the geoid. The unit of geopotential numbers is 1g.p.u.=1kgal·meter=10m2/sec2 according to the unit of acceleration 1gal=1cm/sec2.

A further consequence of (1) is that the gravity vector g at any point P is normal to the equipotential surface, W=WP, through P. Hence, g is tangent to the plumb line through P intersecting normally (by definition) all equipotential surfaces. Let Po denote the intersection point of the plumb line through P and the geoid W=Wo then the orthometric height HP of P is defined by the length of the plumb line section from Po toP. Considering the latter as a path from Po to P it follows from (2) that geopotential number and orthometric height are related as follows, cf. Moritz and Heiskanen (1967), form. (4-19), (4-20), (4-21)

CP = g P HP, HP =

CPg (3)P

where g P (in kgal) is the mean value of gravity g integrated along the plumb line from Po (H=0) to P (H=HP) 1

HP

(4)

g P =P gdH

0

As gravity approx. is 0.98 kgal in the area close to the earth’s surface geopotential numbers in g.p.u. are about 2% smaller than the corresponding orthometric heights in meter.

2. Conversion of leveled height differences.2.1 The inconsistency of leveled heights. According to above the elemental alteration dW of the potential

W corresponding to a vertical elemental displacement dH at the point P is given by dW = -gPdH, cf. Moritz and Heiskanen (1967), form. (2-14). It follows, since gravity g is varying on the equipotential surface through P, that the surface is nonparallel to the neighboring equipotential surfaces. According to chapter 4.1, ibid., the nonparallelism of the equipotential surfaces has the following consequences

▪ leveled height differences are depending on the leveling path between the endpoints▪ summing up leveled height differences along a loop the resulting closing error is not strictly vanishing,

even if the leveling has been performed without any kind of errors▪ the usual observation equation for the adjustment of the leveled height difference from P1 to P2,

Δh12=h2-h1, is not strictly validThat means, leveled heights do not constitute a consistent system. However, orthometric heights or geopotential numbers are free from the lacks above, thus, all leveling can be calculated strictly in the usual

H

1

way, replacing leveled height differences by geopotential or orthometric height differences. From (3) orthometric heights can be computed from geopotential numbers, and vice versa. If metric heights are not needed, say in the computation of loop closing errors, it is often more practical to use geopotential differences.

2.2 Conversion into geopotential differences. As outlined in Moritz and Heiskanen (1967), chapter 4.1., leveled height differences are yielding differences of geopotential numbers (geopotential differences), if combined with gravity. Under normal conditions it holds with sufficient accuracy

ΔC12 = C2 - C1 = Δh12gm (5)cf. Moritz and Heiskanen (1967), form. (4-3), see also the derivations in Torge (1991), sect. 4.3.5, and Jordan/Eggert/Kneissl (1956/1969), § 135. Above Δh12 is the leveled height difference and gm (in kgal) is the arithmetic mean of gravity at the endpoints of the section. Studying the derivations in the textbooks mentioned before it can be seen that equation (5) is valid not only for leveling on the ground, but also for measurements on bridges or in tunnels.

2.3 The orthometric correction. Adding the orthometric correction OC12 to a leveled section height difference Δh12 the corresponding difference of orthometric heights ΔH12=H2-H1 is obtained. Form. (4-33) in Moritz and Heiskanen (1967) is giving the correction in case of leveling on the ground. As can be seen from its derivation it is valid, too, for leveling on bridges or in tunnels. In smaller regions at least, the formula can be written

OC12 = gm go Δhgo

12 g1 go hgo

1 g 2 go hgo2 (6)

Here gm is again the arithmetic mean of gravity at the endpoints, go is an appropriate average value of gravity in the region, g is mean gravity defined through form. (4), and h is a preliminary height from leveling, e.g. Inorder to compute the correction mean gravity gm has to be found from gravity interpolation, cf. sect.5, whereas g is calculated from sect.6. Strictly speaking, form. 6 is the so-called orthometric Helmertcorrection giving Helmert heights when added to leveled height differences.

In case of leveling on the ground above the geoid can be shown that the orthometric Helmert correction referring to ground density ρ=2.00g/cm3 can be simplified as follows

OC12 = -hm(Δg12 + (0.3086 - 2·0.0838)Δh12 )/go (in m)where hm is the arithmetic mean of h1 and h2 (in m) and Δg12=g2-g1 (in mgal) is the difference of gravity at the endpoints. As can be seen from the formula, in a low and flat area with smooth gravity the correction is summing up in the north-south direction, but hardly from east to west. Consequently, its impact on the closing error often is relatively small.

3. Gravity reduction by means of Bouguer plates.Given the points P and Q on the same plumb line gravity reduction from P to Q is the prediction of gravity

at Q from gravity at P. Gravity reduction is needed for the computation of mean gravity g as well as Bougueranomalies ΔgBB used for gravity interpolation. The simplest method is the simplified reduction of Poincaré and Prey, cf. Moritz and Heiskanen (1967), sect.4-3. Though the reduction seems rather rough it is generally working well for leveling in low and flat areas with smooth gravity.

Say, gravity has to be reduced from P to Q. For this purpose the masses along the plumb line through P and Q are replaced by Bouguer plates, i.e. infinite plane horizontal plates with constant mass density, such that both points are on the surface of a such a plate. Assuming P is below Q gravity at Q is obtained according to the following steps:

1) Remove all the plates above P and compute the corresponding gravity at P2) Move P through free air to Q and compute the corresponding gravity at Q3) Restore the plates removed and recomputed the gravity at Q.

An example is given below.Obviously, removing a Bouguer plate located above/below a given point P is increasing/decreasing

gravity at P by an amount corresponding to the plate’s vertical attraction on P, whereas adding a plate above/below the point is decreasing/increasing gravity at P. According to Moritz and Heiskanen (1967), form. (3-15), the attraction is given by

ABB = cρb mgalwhere b (in meter) is the thickness of the plate and cρ is depending on the plate’s mass density ρ (in g/cm3). Thus, the attraction is independent from the distance of P from the plate. Common values of ρ and c ρ are given below:

ground , standard value: ρ=2.67 g/cm3 cρ=0.1119 ground, Denmark: ρ=2.00 cρ=0.0838 sea water: ρ=1.03 cρ=0.0432

1

Note, due to the moraine and chalk densities of the Danish subsurface ground density ρ=2.00 is the traditional value for the computation of leveling in Denmark.

Finally, moving the point P to Q through free air the gravity change is given by F = 0.3086l mgal

where l (in meter) is the distance between P and Q, counted negative if P is below Q.

□ Example: Assuming the point P is below the bottom of the sea (assumed plane) let h denote the height of P and d the depth of the sea. Obviously, since h is negative, t1=-h-d is the positive height of the sea floor above P, cf. case [e] in sect.4. Furthermore, let P1 and Po be the points, where the plumb line through P is intersecting the bottom and the surface of the sea. Now, let s[0,t1] be the distance along the plumb line through P, counted positive from P in upwards direction. Applying the procedure above (ρ=2.00) gravity g(s) at the point corresponding to s is predicted from gravity gP at P by

g (s)=gP+0.0432d+0.0838t1 -0.3086s+0.0838s -0.0838(t1-s)-0.0432d = gP+(2·0.0838-0.3086)s (mgal)Correspondingly, counting the distance from P1, s1=s-t1[0,d] gravity g(s1) is predicted from gravity g1 at P1

g(s1) = g1+0.0432d -0.3086s1 + 0.0432s1 – 0.0432(d-s1) = g1 + (2·0.0432 - 0.3086)s1 (mgal)Hence, applying s=t1 and s1=d gravity at P1 and Po is given by

g1 = gP + (2·0.0838 - 0.3086)t1 (mgal)go = g1 + (2·0.0432 - 0.3086)d (mgal) □

4. Bouguer gravity and anomalies.In order to make gravity gP at a given point P comparable to what is ‘normal’ in accordance with a

geometric/gravimetric model of the earth gravity is reduced from P to the intersection point P o of the plumb line through P and the geoid, assuming a fictitious solid geoid with no masses above. The term ‘solid’ means, e.g., seawater is replaced by soil. The reduced value gBB is the Bouguer gravity of P and its deviation from ‘normal’ is the corresponding Bouguer anomaly ΔgBB

ΔgBB = gBB – γo (7)The ‘normal’ gravity value γo referring to the geodetic reference system GRS80 can be computed from form. (3.4) in Torge (1989), e.g. Obviously, Bouguer anomalies are reflecting the mass densities of the ground, which makes them suitable for gravity interpolation.

In the formulas below, derived in accordance with sect.3 applying ground density ρ=2.00, h is the height of the point P above the geoid/mean sea level and d is the depth of the sea at P’s location. Bouguer gravity g B

is in mgal.[a] P on the ground above geoid:

gB=B gP - 0.0838h + 0.3086h (8)Replacing 0.0838 by 0.1119, cf. sect.3, (8) is corresponding to Moritz and Heiskanen (1967), form. (3-18). Obviously, the Bouguer plate between P and the geoid is removed, hereafter P is lowered to the geoid through free air. Similarly, we find:[b] P at the surface of the sea:

[c] P in free air above the sea: gBB = gP - 0.0432d + 0.0838d (9)

gBB = gP – 0.0432d + 0.0838d + 0.3086h[d] P in free air above the ground above geoid: (ho is the height of P above ground)

gBB = gP - 0.0838(h-ho) + 0.3086h[e] P below the surface of the sea: (t1=-h-d is the height of the bottom above P)

P below the bottom, t1>0: gBB = gP + 0.0838t1 + 0.0432d - 0.3086(-h) + 0.0838(-h)P above the bottom, t1<0: gBB = gP - 0.0432(-t1) + 0.0432(-h) - 0.3086(-h) + 0.0838d

5. Gravity interpolation (GeoGrid)5.1 The GeoGrid program. Gravity interpolation can be carried out by the GeoGrid program, which is part of the program package GRAVSOFT used for geoid computations, cf. Tscherning et al. (1992). Using ground density ρ=2.67 it proceeds as follows.

1. The interpolated Bouguer anomaly ΔgB*B of the point P is computed by the weighted mean of Bougueranomalies of 5 data points in each quadrant, which are closest to P. The weighting is done in accordance with the reciprocal squared distances of P from the data points applied (different options are available).

2. The corresponding Bouguer gravity gB*B is computed according to (7) using normal gravity of GRS80 gB*B = ΔgBB*+ γo

3. Interpolated gravity at P is now computed in two different ways depending on the input height of P.a. input height h>0: P is considered according to case [a] in sect.4, assuming the input height h is the

height of the point considered. Accordingly, gravity g* at the point considered is calculatedanalogously to form. (8) reducing gravity gB*B considered using ground density ρ=2.67g/cm3

from the fictitious geoid, cf. sect.4, to the point

g* = gB*B - 0.3086h + 0.1119h (10)

1

b. input height h<0: P is considered according to case [b], assuming -h is the depth of the sea. Accordingly, gravity g* at the point considered is computed analogously to (9)

g* = gB*B - 0.1119(-h) + 0.0482(-h) (11)Obviously, to make sense the Bouguer anomalies of the data points should refer to ρ=2.67.

5.2 Modifications. For the sake of documentation the GeoGrid program has been used unchanged, however, a few minor modifications would have simplified significantly the following procedure of gravity interpolation.

As mentioned before ρ=2.00 is used for leveling in Denmark. Consequently, the Bouguer anomalies of the data points should refer to the same value. Given anomaly values ΔgBB referring to ρ=2.67 are easily transformed into corresponding values ΔgB referring to ρ=2.00. Regarding a data point P, case [a] or [b], it is immediately seen from formulas corresponding to (7), (8), and (9)

[a]: ΔgB = ΔgBB + (0.1119-0.0838)h (12)The same formula goes for case [b] replacing h in the formula above by -d.

Now, what is happening, if GeoGrid is applying a data list of Bouguer anomalies (ρ=2.00)? Let ΔgB* be the resulting interpolated Bouguer anomaly of the point P and g* the corresponding outcome from GeoGrid. Since the data list is referring to ρ=2.00 GeoGrid should have used the same value in the computation of g*. Taking this into account (modification 1) it is seen from (10) and (11) that no matter of the sign of the input height h the consistent gravity value at the point considered by GeoGrid is

g** = g* + (0.0838-0.1119)h (13)Below, we are giving gravity gP* (in mgal) consistently interpolated based on ρ=2.00 at the point P

depending on the different cases in sect.4. It is assumed the GeoGrid input height of P is the height h of P, except in case [b], where the input height -d is assumed. As is evident from above

[a], [b]: gP* = g**In the other cases further steps are needed, since gravity g** has to be reduced from the point considered

by GeoGrid to the point P (modification 2). Below we are continuing the example in sect.3, i.e. case [e], t1>0.

□ Example (cont.): Since the height of point P is negative, g** is gravity at the point considered, i.e. the point Po, assuming -h is the depth of the sea, cf. 3.b. in sect.5.1. Hence, interpolated gravity gP* at P is obtained removing the fictitious sea water plate from P to Po, lowering Po to P through free air, and adding the plates of soil and water from P to P1 and P1 to Po, respectively. Hence

[e], t1>0: gP* = g** - 0.0432(-h) + 0.3086(-h) - 0.0838t1 - 0.0432d □

Similarly, we find[e], t1<0: gP* = g** - 0.0432(-h) + 0.3086(-h) - 0.0838(-t1) + 0.0432(-t1) - 0.0432(-h) [c]: gP* = g** - 0.0838(h+d) + 0.0432d[d]: gP* = g** - 0.0838ho

6. The computation of mean gravity g .Mean gravity is needed for the transformation of geopotential numbers into orthometric heights, or vice

versa, cf. (3). It is also needed for the computation of orthometric corrections, cf. (6). According to (4) meangravity g P corresponding to the point P is determined from integration of gravity g along the plumb linefrom the geoid to P. However, this gravity is normally unknown. The Helmert approach to solve the problem is to replace the masses along the plumb line by corresponding Bouguer plates. By this, gravity between the geoid and the point P becomes a piecewise linear function of distance in accordance with the different plates. Thus, mean gravity along the section of the plumb line cut out by a given plate, is easily found from the arithmetic mean of gravity at the section’s endpoints.

Corresponding to the different cases in sect.4 we are giving below mean gravity g P (in mgal) based onρ=2.00 of the point P, HP is the orthometric height of P. To make the procedure more comprehensible we are continuing the example in sect.3.

□ Example (cont.): Splitting up the integration according to the plates of soil and sea water between P and Po

mean gravity g P

t1

can be writteng P

HP

= -(I1 + I2)/HP

Here I1 = g(s)ds

0

and I2 = g(s)ds . As is seen from the example g(s) is a piecewise linear function of s int1

accordance with the plates. Thus, I1=½(gP+g1)t1, I2=½(g1+go)d, hence[e], t >0:

g = ½(g P g1 )t1 ½(g1 go )d

1 PPH

1

GeoGrid

(mod.1)

(mod.1)

(mod.2)

gravity g*[c]-[e]

interpolated case [a]interpol. points and input heights

Bouguer anomaliesΔgB (2.00)

mod. gravity g**

orthom. height diff.

adj. geopot. numbersadj. Helmert heights

orthom. corrections

geopotential differenceslev. height diff.

final gravity gP*mean gravity g P

Gravity gP is often found by interpolation, whereas go and g1 are given in sect.3. □Similarly, mean gravity corresponding to P in all other cases is given by

[e], t1<0:[a]:[c] :

g P = gP + ½(0.3086 - 2·0.0432)Hp

g P = gP + ½(0.3086 - 2·0.0838)Hp

g P = gP + ½·0.3086Hp

[d] :g =

½(g o g1 )(H P ho ) ½(g1 g P )hoP HP

where go is gravity at the geoid below P and g1 is corresponding to the ground , i.e.go = g1 + (0.3086 - 2·0.0838)(HP - ho) g1 = gP + 0.3086ho

7. Dataflow. The diagram below is illustrating the entire dataflow during gravity interpolation and leveling adjustment. The dot-and-dash lines are irrelevant, since geopotential differences have not been used. The denotation is in accordance with previous sections.

transform.

.

.

.

datapoints: Bouguer ano- malies ΔgB

1

Appendix 21. Stability investigation in the Helsingør region

According to the set up in table 1 we have to look for a stable subsoil benchmark leveled in 1980 and 1992. Since most of the subsoil points have been omitted in the leveling from 1992 the number of candidates can be narrowed down to the following: Kvistgård (G.I.1646, G.I.1647, G.I.1648), Hamlets Grav (G.I.1605, G.I.1606, G.I.1607), and G.M.1341. The point last-mentioned is unsuitable for connection, it is located on a bastion of the Kronborg Castle and a subsidence of about 35mm relative to Hamlets Grav has taken place from 1940 to 1992. Due to numerous surveying campaigns in the area precise leveling has been conducted frequently: 1898 (First Precise Leveling), 1908, 1939, 1940, 1941, 1942, 1943 (Second Precise Leveling), 1980, 1983, and 1992 (Third Precise Leveling). However, in order to evaluate stability between and within the groups Hamlets Grav and Kvistgård, cf. fig. 4, only the levelings from 1941 on are relevant, simply because the groups did not exist before 1940 and 1941, respectively. The data sources applied are indicated in table A1, where the pages are referring to computation volumes of G.I. For the sake of completeness we also give the references of the oldest levelings: 1898(#DK_niDprs), 1908(p.142 ff.), 1939(p.2058 ff.).

1.1 Leveling within the groups of Hamlets Grav and Kvistgård. In table A1 we have listed all measurements available. The results recorded are means of 4 to 9 single runs with sight lengths of about 35m.

Hamlets Grav: 1940 1941 1942 1943 1980 1983 1992G.I.1605

-0.55719 -0.55692 -0.55673 -0.55652 -0.55799 -0.55809 -0.55776G.I.1606

-0.89963 -0.89914 -0.89946 -0.89966 -0.89785 -0.89804 -0.89827G.I.1607 ----------- ----------- ----------- ----------- ----------- ----------- -----------

-1.45682 -1.45606 -1.45619 -1.45618 -1.45584 -1.45613 -1.45603source: p.2077 ff. p.2186 ff p.2366 ff p.7709 ff #DK_niDniv #DK_niDniv #DK_niDprs

--------------------------------------------------------------------------------------------------------------------------------------------Kvistgård: 1941 1942 1943 1980 1983 1992

G.I.1646-0.96802 -0.96806 -0.96804 -0.96890 -0.96902 -0.96894

G.I.1647-0.75873 -0.75860 -0.75872 -0.75801 -0.75786 -0.75654

G.I.1648 ----------- ----------- ----------- ----------- ----------- ------------1.72675 -1.72666 -1.72676 -1.72691 -1.72688 -1.72548

source: #DK_niDprs #DK_niDprs #DK_niDprs #DK_niDniv #DK_niDniv #DK_niDprs

table A1: Repeated levelings within the groups

Considering the group Hamlets Grav we can conclude that G.I.1605 and G.I.1607 can be considered mutually stable from 1941 to 1992, whereas G.I.1606 has subsided about 1½mm during the years from 1943 to 1980, hereafter the point seems stable. That means any one of the three benchmarks could be used for the connection 1980/1992. As regards the Kvistgård group we notice mutual stability of G.I.1646 and G.I.1647 from 1980 on, whereas G.I.1648 has raised about 1½mm (according to the field books this is probably caused by some construction work close to the point).

1.2 Levelling between the groups. Below, we are giving the levelings available from Kvistgård (G.I.1647) to Hamlets Grav (G.I.1607), which are including the old subsoil benchmark G.M.1337 from the First Precise Leveling. The results recorded are rounded measured height differences summed up along the shortest route connecting the groups. The same data sources as above have been used.

1940 1941 1942 1943 1980 1983 1992G.I.16474.5km -4.513 -4.516 -4.515 -4.517 -4.516 -

G.M.13374.5km -16.278 -16.276 -16.279 -16.277 -16.267 -16.272 -

G.I.1607 ---------- ---------- ----------- ---------- -----------20.790 -20.794 -20.792 -20.784 -20.788 -20.789

table A2: Repeated levelings between the groups

From the table we can conclude that G.I.1647 and G.M.1337 seem mutually stable from the forties up to the eighties, whereas in 1980 there is an apparent raise of G.I.1607 of about 10mm relative to G.M.1337. Since this could not be explained neither by a gross leveling error nor by postglacial uplift a new leveling was carried out in 1983, now indicating a subsidence of about 5mm since 1980. The validity of these contradicting results seems rather unlikely. However, since the result from 1983 is confirmed in 1992 we have decided to ignore the leveling from 1980 as much as possible. This is excluding the Kvistgård benchmarks for the connection 1980/1992, thus, the benchmark G.I.1607, which is directly entering the leveling line, has been selected as connection point. As is seen from the table there might be a raise of a few

1

millimeters of G.I.1607 relative to G.I.1647 during the period from the forties up to the nineties. This is in good agreement with the postglacial land uplift according to Ekman (1996).

The reason for the suspicious deviation of the height difference in 1980 from G.M.1337 and G.I.1607 can hardly be found today, but perhaps it should be mentioned that most of the leveling sections in 1980 were measured by three single runs according to the field book GA.I, journal bind 102. That means systematic errors might not be fully removed from the means, which have been used in the computation of the height difference in question.

2. Gravity along the Link.2.1 Gravity and Bouger anomalies of the new gravity points, ρ=2.00. The gravity profile of the Link

has been measured relative to the KMS gravity station 120 in the beginning of April 2000 by two LaCoste Romberg gravimeters, G867 and G466. The location of the new points can be seen from the bold points in fig.7. Measured gravity gP and corresponding Bouguer anomalies ΔgB computed from the formulas in app.1, sect.4, are given below. Normal gravity γo has been computed as mentioned in app.1, sect.4. The leveled heights h of the points in the tunnel are including an offset of 65 cm, since gravity was measured at the bottom of tunnel. The depth d of the sea has been estimated roughly from a marine map.

point no. loc. gP h d ΔgB

4078/432 661 EastBridge 981 531.40 mgal 26.84 m 2 m -15.76 mgal4078/430 661 East Bridge 521.17 61.71 7 -15.504078/411 061 West Bridge 524.65 53.46 7 -15.364078/301 119 Peberholm 539.63 12.34 - -15.144078/300 501 Tunnel 543.93 -3.70 3 -15.904078/919 Tunnel 547.19 -19.59 9 -16.24

table A3: Measured gravity and Bouguer anomalies (ρ=2.00) of the gravity profile of the Link

2.2 Interpolated gravity gP*, ρ=2.00. The gravity values in table A4 have been computed according to app.1, sect 5.2. Bold point numbers are referring to stable points kept in the KMS point database. The last column is indicating points, which have been measured by one national team only.

point P h d gP*Bridge 4078/500 503 19.48 m 2 m 981533.88 mgal

4078/500 502 21.99 2 981532.95 Sw4078/500 501 22.00 2 981532.974078/500 501.1 22.00 2 981532.97 DK4078/432 861 23.86 2 981532.33 Sw4078/432 661 26.84 2 981531.404078/432 461 30.34 2 981530.364078/432 261 33.81 2 981529.344078/432 061 37.35 3 981528.24 DK4078/431 961 37.40 3 981528.23 Sw4078/431 861 40.82 3 981527.224078/431 661 44.26 4 981526.184078/431 461 47.76 5 981525.124078/431 261 51.22 5 981524.184078/431 061 54.75 6 981523.164078/430 861 58.22 6 981522.204078/430 661 61.71 7 981521.174078/430 461 64.78 7 981520.284078/430 261 67.07 7 981519.57

(pylon) 4078/400 506 68.88 7 981518.94(pylon) 4078/400 502 68.91 7 981518.71

4078/410 261 67.13 7 981519.154078/410 461 64.78 7 981519.774078/410 661 61.70 7 981521.014078/410 861 57.82 7 981522.954078/411 061 53.46 7 981524.654078/411 261 49.03 7 981526.024078/411 461 44.66 6 981527.424078/411 661 40.25 6 981528.834078/411 861 35.88 5 981530.344078/412 061 31.48 5 981531.824078/412 261 27.74 5 981533.134078/412 461 23.94 5 981534.444078/413 261 20.28 4 981535.78 Sw

Bridge 4078/300 511 15.63 4 981537.51Peberholm 4078/301 119 12.34 - 981539.63

4078/301 047 11.40 - 981539.954078/301 011 11.38 - 981540.01

Peberholm 4078/300 975 9.78 - 981540.48Tunnel 4078/300 502 -0.64 3 981543.29

4078/300 501 -3.05 3 981543.85 (to be continued)

1

(continued)4078/1 409 -9.56 3 981544.934078/1 919 -14.61 5 981545.914078/1 719 -16.30 5 981546.314078/1 519 -17.31 9 981546.774078/1 319 -18.38 9 981547.084078/1 119 -19.44 9 981547.474078/919 -18.94 9 981547.094078/719 -17.90 9 981547.164078/519 -16.86 9 981546.964078/319 -15.50 5 981546.774078/119 -10.78 3 981546.134078/100 502 -4.61 2 981545.30

Tunnel 4078/100 501 -1.80 2 981544.89

table A4: Interpolated gravity (ρ=2.00) along the Link

●●●

References

Abramovitz , T. et al. (1997): Proterozoic sutures and terranes in the southeastern Baltic Shield interpreted from BABEL deep seismic data. Tectonophysics 270, pp. 259-277

Becker, J.-M. (1987): The experiences with new leveling techniques ML and MTL. In: Pelzer, H., Niemeyer, W. (1987): Determination of Heights and Height Changes. Ferd. Dümmlers Verlag, Bonn

Becker, J.-M., Bedsted Andersen, O. (1986): Guidelines for motorized 1. order precise leveling. In: Proceedings of the 10th General Meeting of the Nordic Geodetic Commission 1986. Finnish Geodetic Institute, Helsinki

Bedsted Andersen, O. et al. (1986): Water Crossing Leveling between Denmark and Sweden 1980-1981. Geodætisk Instituts Skrifter 3.Række, Bind XLV

Blakely, R.J. (1996):Potential Theory in Gravity and Magnetic Applications. Cambridge University Press Den Danske Gradmaaling (1909): Nivellement over bredere Vandarealer. Ny Række, Hefte 4, Copenhagen Ekman, M. (1996): A consistent map of the postglacial uplift of Fennoscandia. Terra Nova 8,158-165Heiskanen, A., Moritz, H. (1967): Physical Geodesy. W.H. Freeman and Company, San Francisco and London Jordan/Eggert/Kneissl (1956/1969): Handbuch der Vermessungskunde, Band V. J.B. Metzlersche Verlagsbuchhandlung, Stuttgart Nørlund, N.E. (1946): Hydrostatisk Nivellement over Øresund. Geodætisk Instituts Skrifter 3. Række, Bind VIIISchmidt, K. (2000): The new Danish Height system DVR90. Publ.s, 4.series, volume 8. The National Survey and Cadastre, Copenhagen Tscherning, C.C., Forsberg, R., Knudsen, P. (1992): The GRAVSOFT package for geoid determination. Proc. of the 1st continental

workshop on the geoid in Europe, pp. 327-334. Prague Torge, W. (1991): Geodesy. Second edition. Walter de Gruyter, Berlin, New York Torge, W. (1989): Gravimetry. Walter de Gruyter, Berlin, New York

1

Hamlets Grav - Pålsjø CliffDouble runs of the Oeresund Loop

G.I.1607K -06-09006

K -06-09006K -06-09034

555430

-18.400510.99555

022*1*5511 022*2*5505022*2*5505 022*1*5512

1052886

-2.058223.24222

K -06-09034 K -06-00087 344 -2.91636 022*1*5512 022*2*4501 685 0.32640K -06-00087 K -06-00053 518 -0.46586 022*2*4501 022*1*4505 958 -3.45540K -06-00087 K -06-00053 534 -0.46664 022*1*4505 022*1*4506 1170 3.48181K -06-00053 K -06-09168 217 2.55088 022*1*4506 022*1*4507 1248 -2.04387K -06-09168 K -06-09008 227 4.57693 022*1*4507 022*1*4508 924 -1.22395K -06-09008 K -06-09170 22 21.7468 022*1*4508 022*2*4504 1272 -0.02258K -06-09170 032*2*3116 4800 -0.5903 022*2*4504 022*1*4509 816 -1.29970Pålsjø Cliff - East Bridge 022*1*4509 022*1280*0764 162 1.40626032*2*3116 032*2*3102 476 -26.42334 022*1280*0764 022*1280*0959 455 0.21849032*2*3102 032*1283*0713 197 2.57716 022*1280*0959 022*1*3506 419 0.31138032*1283*0713 032*1283*0953 880 -2.74134 022*1*3506 022*1280*0958 872 -1.12392032*1283*0953 032*1283*0922 778 -0.80837 022*1280*0958 022*1*3507 859 3.02548032*1283*0922 032*1283*0703 361 4.80158 022*1*3507 022*1280*0080 300 0.64929032*1283*0703 032*1283*1032 250 -0.47496 022*1280*0080 022*1*3508 336 -0.12420032*1283*1032 032*1*3124 196 25.30183 022*1*3508 022*1280*0288 379 0.63550032*1*3124 032*1*3119 295 -0.23249 022*1280*0288 022*1*3509 265 -0.81723032*1*3119 032*1*3120 1189 -7.57831 022*1*3509 022*1280*0141 401 -1.10018032*1*3120 032*1*3121 965 -16.81553 022*1280*0141 022*1*3510 431 0.66180032*1*3121 032*9999*3121 429 -0.36191 022*1*3510 022*1280*0142 244 2.50207032*9999*3121 032*1*3123 1418 -0.26218 022*1280*0142 022*1*3511 460 -0.12100032*1*3123 032*2*2101 1050 1.60767 022*1*3511 022*1280*0255 164 1.26987032*2*2101 032*1*2104 1287 -3.72564 022*1280*0255 022*1*3401 591 0.45717032*1*2104 032*1*2105 941 0.31725 022*1*3401 022*1280*0249 385 2.45994032*1*2105 032*1*2106 830 -2.86912 022*1280*0249 022*1*3402 487 0.01928032*1*2106 032*1*2107 1167 19.63731 022*1*3402 022*1280*0465 643 2.87132032*1*2107 032*1*2206 624 -2.24019 022*1280*0465 022*1*2425 464 1.85279032*1*2206 032*1*1204 1569 -1.47465 022*1*2425 022*1280*0956 435 -2.65869032*1*1204 032*1*1210 1047 2.94908 022*1280*0956 022*2*2414 642 -1.37452032*1*1210 032*1*1211 1432 14.73561 022*2*2414 022*1280*0477 525 0.40207032*1*1211 032*1*1212 1103 2.72038 022*1280*0477 022*1280*0914 193 -0.01146032*1*1212 032*1*1213 523 6.02802 022*1280*0914 022*1*2422 388 4.15367032*1*1213 032*1*1309 1009 7.69969 022*1*2422 022*1*2421 675 -0.64702032*1*1309 032*11*1317 576 -9.98954 022*1*2421 022*1*2420 813 9.48434032*11*1317 032*1*1311 1687 8.36616 022*1*2420 022*1*2419 556 -2.03922032*1*1311 032*1*1316 660 1.66891 022*1*2419 022*1*2418 913 -1.84510032*1*1316 032*11*0309 1081 9.63501 022*1*2418 022*1*2417 1286 -3.84499032*11*0309 032*1*0205 1084 -5.36118 022*1*2417 022*1*2423 939 -1.86249032*1*0205 032*1*0206 1048 -10.48983 022*1*2423 022*9999*2423 492 -2.89425032*1*0206 032*11*0210 1268 -16.94041 022*9999*2423 022*46*200004 756 2.04942032*11*0210 032*1*0209 1155 -17.83328 022*46*200004 022*1*2311 857 -1.78011032*1*0209 032*1*0207 592 0.01595 022*1*2311 022*1*2310 818 0.62461032*1*0207 022*1*9204 1236 -2.78429 022*1*2310 022*46*200003 648 -0.70353022*1*9204 022*1*9205 1236 -5.30230 022*46*200003 022*1*2309 367 -3.60825022*1*9205 022*1*9309 1334 -1.55838 022*1*2309 022*1*2308 916 0.21854022*1*9309 022*1*9310 1031 0.71776 022*1*2308 022*46*200001 609 -0.38980022*1*9310 022*9999*9304 1531 0.83602 022*46*200001 4078/500 503 978 9.96653022*9999*9304 022*1*9312 1294 -1.41624 East Bridge - High Bridge022*1*9312 022*1*9313 898 5.43764 4078/500 503 4078/500 501 225 2.52102022*1*9313 022*1*8303 997 -5.72160 4078/500 503 4078/500 501.1 274 2.51802022*1*8303 022*1*8304 813 5.73844 4078/500 502 4078/500 501 81 0.00201022*1*8304 022*1*8305 1318 -4.60347 4078/500 501 4078/432 661 389 4.83744022*1*8305 022*1*8306 1278 0.59006 4078/500 501.1 4078/432 661 394 4.84160022*1*8306 022*1*8409 790 -0.31506 4078/432 661 4078/432 461 281 3.50224022*1*8409 022*1*7409 1887 0.09927 4078/432 661 4078/432 461 279 3.50303022*1*7409 022*1*7408 982 -0.57922 4078/432 461 4078/432 261 280 3.46916022*1*7408 022*1*7303 1111 0.40551 4078/432 461 4078/432 261 278 3.46932022*1*7303 022*1*7301 1048 -0.17979 4078/432 261 4078/432 061 277 3.53847022*1*7301 022*9999*7410 714 7.87950 4078/432 261 4078/432 061 275 3.53928022*9999*7410 022*1*7403 1368 -4.45244 4078/432 061 4078/431 861 282 3.46901022*1*7403 022*1*7404 1079 -1.82808 4078/432 061 4078/431 861 281 3.46984022*1*7404 022*1*6406 1344 6.07948 4078/431 861 4078/431 661 278 3.44458022*1*6406 022*1*6405 1095 5.42284 4078/431 661 4078/431 461 279 3.50287022*1*6405 022*11*6407 1253 -2.81875 4078/431 661 4078/431 461 280 3.50370022*11*6407 022*1*6517 989 -5.87114 4078/431 461 4078/431 261 281 3.45594022*11*6407 022*1*6517 987 -5.87151 4078/431 461 4078/431 261 281 3.45666022*1*6517 022*1*6518 1298 2.72412 4078/431 261 4078/431 061 283 3.52902022*1*6517 022*1*6518 1290 2.72445 4078/431 261 4078/431 061 280 3.52872022*1*6518 022*1*6519 1401 4.56417 4078/431 061 4078/430 861 280 3.46970022*1*6519 022*1*6515 799 2.28259 4078/431 061 4078/430 861 280 3.47021022*1*6515 022*1*6516 1292 -11.75967 4078/430 861 4078/430 661 279 3.48958022*1*6516 022*1*5507 1266 4.07845 4078/430 861 4078/430 661 281 3.48961022*1*5507 022*1*5508 1261 -4.85728 4078/430 661 4078/430 461 279 3.06753022*1*5508 022*1*5509 1204 -1.55751 4078/430 661 4078/430 461 279 3.06866022*1*5509 022*1*5510 1247 0.17462 4078/430 461 4078/430 261 276 2.29430022*1*5510 022*1*5511 1228 0.83019 4078/500 503 4078/500 502 227 2.51847

2

4078/500 503 4078/500 501 224 2.52099 4078/410 861 4078/411 061 282 -4.359104078/500 503 4078/500 501.1 224 2.51856 4078/411 061 4078/411 261 283 -4.426474078/500 501 4078/432 861 159 1.86373 4078/411 061 4078/411 261 283 -4.426154078/500 501.1 4078/432 861 148 1.86723 4078/411 061 4078/411 461 564 -8.792874078/432 861 4078/432 661 242 2.97255 4078/411 261 4078/411 461 282 -4.365984078/432 861 4078/432 661 253 2.97289 4078/411 261 4078/411 461 282 -4.365384078/432 661 4078/432 461 282 3.50221 4078/411 461 4078/411 661 282 -4.409224078/432 661 4078/432 461 282 3.50208 4078/411 461 4078/411 661 282 -4.408654078/432 461 4078/432 261 282 3.46879 4078/411 461 4078/411 861 563 -8.782344078/432 261 4078/431 961 285 3.59686 4078/411 461 4078/411 861 563 -8.783784078/432 261 4078/431 961 285 3.59670 4078/411 661 4078/411 861 280 -4.373744078/431 961 4078/431 861 280 3.41143 4078/411 661 4078/411 861 281 -4.373354078/431 961 4078/431 861 280 3.41133 4078/411 861 4078/412 061 282 -4.398054078/431 861 4078/431 661 282 3.44495 4078/411 861 4078/412 061 282 -4.396944078/431 861 4078/431 661 282 3.44450 4078/411 861 4078/412 261 524 -8.134614078/431 661 4078/431 461 282 3.50264 4078/411 861 4078/412 261 523 -8.134874078/431 661 4078/431 461 282 3.50300 4078/412 061 4078/412 261 242 -3.737274078/431 661 4078/431 261 563 6.95879 4078/412 061 4078/412 261 242 -3.736954078/431 461 4078/431 261 282 3.45670 4078/412 261 4078/412 461 245 -3.800094078/431 461 4078/431 261 282 3.45619 4078/412 261 4078/412 461 245 -3.799524078/431 261 4078/431 061 284 3.52845 4078/412 261 4078/413 261 479 -7.464414078/431 261 4078/431 061 283 3.52812 4078/412 261 4078/413 261 479 -7.464704078/431 261 4078/430 861 564 6.99879 4078/412 461 4078/413 261 234 -3.664534078/431 061 4078/430 861 282 3.47004 4078/412 461 4078/413 261 250 -3.663884078/431 061 4078/430 861 282 3.47036 4078/413 261 4078/300 511 347 -4.650544078/430 861 4078/430 661 282 3.48938 4078/413 261 4078/300 511 347 -4.651124078/4304078/430

861861

4078/4304078/430

661461

282563

3.489556.55795

4078/413 261 4078/300Peberholm - Tunnel

511 349 -4.65219

4078/430 661 4078/430 461 282 3.06779 4078/300 511 4078/301 119 482 -3.283644078/430 461 4078/430 261 353 2.29429 4078/301 119 4078/301 083 478 -0.962984078/430 461 4078/430 261High Bridge - West Bridge

280 2.29449 4078/3014078/301

119083

4078/3014078/301

083047

479481

-0.963150.01445

4078/430 261 4078/400 506 306 1.80896 4078/301 083 4078/301 047 481 0.014444078/430 261 4078/400 506 303 1.80869 4078/301 047 4078/301 011 478 -0.015564078/400 506 4078/400 502 529 0.02962 4078/301 047 4078/301 011 482 -0.015674078/400 506 4078/400 502 490 0.02833 4078/301 011 4078/300 975 481 -1.602874078/400 502 4078/410 261 434 -1.77911 4078/301 011 4078/300 975 479 -1.603114078/400 502 4078/410 261 301 -1.77923 4078/300 975 4078/300 502 669 -10.416394078/430 261 4078/400 506 307 1.80986 4078/300 975 4078/300 502 668 -10.416404078/430 261 4078/400 506 306 1.80885 4078/300 502 4078/300 501 92 -2.414404078/430 261 4078/400 506 307 1.81112 4078/300 502 4078/300 501 96 -2.413904078/400 506 4078/400 502 495 0.02905 4078/300 511 4078/301 119 482 -3.282774078/400 502 4078/410 261 304 -1.77968 4078/300 511 4078/301 119 484 -3.283574078/400 502 4078/410 261 304 -1.78008 4078/301 119 4078/301 083 481 -0.963994078/400 502 4078/410 261 305 -1.77994 4078/301 119 4078/301 083 482 -0.963734078/400 502 4078/410 261 304 -1.77933 4078/301 083 4078/301 047 485 0.014084078/400 502 4078/410West Bridge - Peberholm

261 304 -1.77955 4078/3014078/301

083083

4078/3014078/301

047047

482481

0.014380.01445

4078/410 261 4078/410 461 278 -2.35130 4078/301 047 4078/301 011 493 -0.016704078/410 261 4078/410 461 278 -2.35010 4078/301 047 4078/301 011 481 -0.016304078/410 461 4078/410 661 277 -3.07664 4078/301 047 4078/301 011 482 -0.016744078/410 661 4078/410 861 279 -3.88796 4078/301 011 4078/300 975 484 -1.603124078/410 661 4078/410 861 281 -3.88824 4078/301 011 4078/300 975 482 -1.603094078/410 861 4078/411 061 280 -4.35990 4078/300 975 4078/300 502 676 -10.417394078/410 861 4078/411 061 281 -4.35970 4078/300 975 4078/300 502 668 -10.417214078/411 061 4078/411 261 282 -4.42618 4078/300 502 4078/300 501 99 -2.414054078/411 061 4078/411 261 283 -4.42741 4078/300 502 4078/300 501 94 -2.413374078/411 261 4078/411 461 281 -4.36659 4078/300 502 4078/300 501 95 -2.413754078/411 261 4078/411 461 279 -4.36644 4078/300 502 4078/300 501 96 -2.414064078/4114078/411

461461

4078/4114078/411

661661

283280

-4.40889-4.40904

4078/300Tunnel -

502 4078/300Copenhagen

501 94 -2.41392

4078/411 661 4078/411 861 283 -4.37433 4078/300 501 4078/1 409 331 -6.501294078/411 661 4078/411 861 280 -4.37394 4078/300 501 4078/1 409 272 -6.501144078/411 861 4078/412 061 282 -4.39739 4078/1 409 4078/1 919 356 -5.050404078/411 861 4078/412 061 279 -4.39711 4078/1 409 4078/1 919 357 -5.050124078/412 061 4078/412 261 244 -3.73762 4078/1 919 4078/1 719 350 -1.686944078/412 061 4078/412 261 239 -3.73690 4078/1 919 4078/1 719 351 -1.686504078/412 261 4078/412 461 245 -3.80004 4078/1 719 4078/1 519 352 -1.016084078/412 261 4078/412 461 245 -3.80019 4078/1 719 4078/1 519 352 -1.016544078/412 461 4078/300 511 581 -8.31633 4078/1 519 4078/1 319 348 -1.073004078/410 261 4078/410 461 280 -2.35134 4078/1 519 4078/1 319 349 -1.073184078/410 261 4078/410 461 280 -2.34970 4078/1 319 4078/1 119 352 -1.062284078/410 261 4078/410 661 564 -5.42706 4078/1 319 4078/1 119 351 -1.061854078/410 261 4078/410 661 561 -5.42764 4078/1 119 4078/919 350 0.500494078/410 461 4078/410 661 282 -3.07612 4078/1 119 4078/919 350 0.500244078/410 461 4078/410 661 282 -3.07555 4078/919 4078/719 348 1.042034078/410 661 4078/410 861 282 -3.88833 4078/919 4078/719 350 1.041684078/410 661 4078/410 861 281 -3.88822 4078/719 4078/519 347 1.044124078/410 661 4078/411 061 563 -8.24807 4078/719 4078/519 349 1.044004078/410 861 4078/411 061 282 -4.35942 4078/519 4078/319 348 1.35837

2

4078/519 4078/319 351 1.35806 K -01-06429 K -01-08057 290 -0.644724078/319 4078/119 349 4.71790 K -01-08057 K -01-08175 225 0.492464078/319 4078/119 352 4.71824 K -01-08175 K -01-06400 150 0.102734078/119 4078/100 502 351 6.17122 K -01-06400 K -01-06382 336 -0.965484078/119 4078/100 502 343 6.17069 K -01-06382 K -01-06346 298 1.128424078/100 502 4078/100 501 107 2.80724 K -01-06346 K -01-06335 207 -0.188274078/100 502 4078/100 501 108 2.80674 K -01-06335 K -01-09038 172 -0.203414078/300 501 4078/1 409 274 -6.50110 K -01-09038 K -01-09037 254 -0.228164078/300 501 4078/1 409 268 -6.50124 K -01-09037 K -01-06272 411 0.993714078/1 409 4078/1 919 358 -5.05049 K -01-06272 K -01-06261 288 0.893644078/1 409 4078/1 919 358 -5.05047 K -01-06261 K -01-07657 559 1.364494078/1 409 4078/1 919 359 -5.05054 K -01-07657 K -01-06682 277 0.349014078/1 919 4078/1 719 353 -1.68648 K -01-06682 K -01-06179 304 -1.308604078/1 919 4078/1 719 353 -1.68687 K -01-06179 K -01-08018 331 -1.338614078/1 919 4078/1 719 352 -1.68687 K -01-08018 K -01-06799 445 -0.088854078/1 719 4078/1 519 354 -1.01576 K -01-06799 K -01-06109 379 1.759794078/1 719 4078/1 519 353 -1.01575 K -01-06109 G.M.1385/1386 186 1.138164078/1 719 4078/1 519 354 -1.01574 G.M.1385/1386 K -01-06110 255 1.523284078/1 519 4078/1 319 351 -1.07299 K -01-06110 G.M.1384/1385.1 150 1.195684078/1 519 4078/1 319 351 -1.07298 G.M.1384/1385.1 K -01-06111 28 0.585844078/1 519 4078/1 319 350 -1.07330 K -01-06111 K -01-07444 326 -0.304294078/1 319 4078/1 119 352 -1.06147 K -01-07444 K -01-07446 404 2.327514078/1 319 4078/1 119 352 -1.06183 K -01-07446 K -01-06802 427 1.084504078/1 319 4078/1 119 352 -1.06210 K -01-06802 K -01-07593 294 0.048734078/1 119 4078/919 353 0.50079 K -01-07593 K -01-07374 119 -0.444804078/1 119 4078/919 353 0.50024 K -01-07374 K -01-07592 329 0.189624078/1 119 4078/919 352 0.50042 K -01-07592 K -01-08007 339 -1.847514078/919 4078/719 351 1.04180 K -01-08007 K -01-08070 578 -3.487484078/919 4078/719 352 1.04195 K -01-08070 K -01-07536 232 3.408254078/919 4078/719 351 1.04157 K -01-07536 K -01-06024 198 5.733504078/719 4078/519 352 1.04383 K -01-06024 K -01-08191 362 1.848734078/719 4078/519 351 1.04365 K -01-08191 K -01-07535 340 0.709984078/719 4078/519 352 1.04360 K -01-07535 K -01-07103 590 3.288404078/519 4078/319 351 1.35766 K -01-07103 K -01-06876 278 -0.048204078/519 4078/319 351 1.35772 K -01-06876 K -01-07522 306 0.334394078/519 4078/319 352 1.35742 K -01-07522 K -01-07106 183 0.745564078/319 4078/119 352 4.71838 K -01-07106 K -01-06981 296 -0.065084078/319 4078/119 352 4.71759 K -01-06981 K -01-07594 315 -3.539024078/319 4078/119 352 4.71776 K -01-07594 1-13-09047 590 3.925984078/119 4078/100 502 290 6.17035 1-13-09047 1-13-09046 180 2.833774078/119 4078/100 502 288 6.16971 1-13-09046 1-13-09045 276 4.129914078/119 4078/100 502 289 6.17006 1-13-09045 1-13-09036 480 10.125744078/100 502 4078/100 501 107 2.80696 1-13-09036 1-13-09035 375 4.974334078/100 502 4078/100 501 106 2.80704 1-13-09035 1-13-09085 165 4.458714078/100 502 4078/100 501 107 2.80691 1-13-09085 1-13-09015 271 -0.22734Copenhagen - Hamlets Grav 1-13-09015 G.I.1806 42 -3.352334078/100 501 92 164 807 5.43994 G.I.1806 G.I.1805 33 -0.698594078/100 501 92 164 811 5.43855 G.I.1806 G.I.1805 33 -0.6986292 164 1-06-09129 608 1.25525 G.I.1805 1-02-09016 86 -0.5287992 164 1-06-09129 607 1.25497 1-02-09016 1-02-09154 246 -3.348911-06-09129 1-06-09009 372 -2.84618 1-02-09154 1-02-09235 371 -4.697361-06-09129 1-06-09009 300 -2.84556 1-02-09235 1-02-09027 844 -4.960561-06-09129 1-06-09009 345 -2.84511 1-02-09027 1-02-09197 575 0.400661-06-09009 1-06-09130 322 0.29346 1-02-09197 1-02-09023 186 1.842841-06-09009 1-06-09130 329 0.29414 1-02-09023 1-02-09020 741 -3.768281-06-09009 1-06-09130 313 0.29329 1-02-09020 1-07-09059 364 0.807261-06-09130 1-06-09030 458 -0.42906 1-07-09059 1-07-09060 395 -1.359581-06-09130 1-06-09030 437 -0.42810 1-07-09060 1-07-09034 1271 -10.358941-06-09130 1-06-09030 410 -0.42850 1-07-09034 1-07-09005 278 1.131951-06-09030 1-06-09029 262 0.28163 1-07-09005 1-07-09004 956 13.254871-06-09030 1-06-09029 245 0.28100 1-07-09004 1-07-09061 199 -0.129451-06-09029 K -01-08160 939 -0.40423 1-07-09061 1-07-09062 423 2.01512K -01-08160 K -01-07231 322 -0.82047 1-07-09062 1-07-09063 608 -0.32514K -01-07231 K -01-07675 300 1.66577 1-07-09063 1-07-09066 205 -1.32782K -01-07675 K -01-07736 431 -1.27456 1-07-09066 1-07-09064 219 -2.00371K -01-07736 K -01-07800 342 1.21947 1-07-09064 1-07-09065 463 -4.81246K -01-07800 K -01-07735 417 -0.45409 1-07-09065 1-14-09047 1122 21.27404K -01-07735 K -01-07805 436 0.02328 1-14-09047 1-14-09048 640 -23.19757K -01-07805 K -01-07833 373 -0.35008 1-14-09048 1-14-09003 668 15.81157K -01-07833 K -01-07225 680 -0.86047 1-14-09003 G.I.2097 618 11.33078K -01-07225 K -01-07733 262 1.89191 G.I.2097 1-14-09012 529 -0.63682K -01-07733 K -01-07825 384 -0.60881 1-14-09012 1-14-09014 456 4.38710K -01-07825 K -01-07559 507 0.17650 1-14-09014 13-01-09175 979 -4.89642K -01-07559 K -01-08065 773 0.02748 13-01-09175 13-01-09176 869 -6.46304K -01-08065 K -01-07769 612 0.12699 13-01-09176 13-01-09035 788 -0.49027K -01-07769 K -01-08163 192 -0.07035 13-01-09035 13-01-09042 320 -3.92737K -01-08163 K -01-06475 93 0.00546 13-01-09042 13-01-09055 403 0.40165K -01-06475 K -01-07438 205 0.44600 13-01-09055 13-01-09155 1170 2.10255K -01-07438 K -01-06817 612 0.78750 13-01-09155 13-01-09079 507 6.73383K -01-06817 K -01-06429 193 -0.54556 13-01-09079 13-01-09157 764 -9.16765

2

13-01-09157 13-01-09178 722 9.1374513-01-09178 13-01-09152 1055 -21.7658413-01-09152 13-01-09068 895 7.7813513-01-09068 13-03-09002 896 -4.2245513-03-09002 13-03-09001 464 -2.7151913-03-09001 13-05-09079 572 -3.3771913-05-09079 13-05-09070 776 -1.5931013-05-09070 13-05-00802 812 -0.2750513-05-00802 13-05-09001 1226 -7.1941913-05-09001 13-04-09010 1069 6.0779413-04-09010 G.M.1330 483 0.18952G.M.1330 13-04-09083 146 -0.6426713-04-09083 13-04-09026 836 -2.6581113-04-09026 13-04-09006 893 -16.9166513-04-09006 13-04-09004 737 20.4872213-04-09004 G.M.1332 938 -8.08294G.M.1332 12-01-09154 1203 1.0593012-01-09154 12-01-09155 1279 14.6937512-01-09155 12-01-09156 919 11.3636812-01-09156 G.M.1334 822 7.37570G.M.1334 12-02-09010 722 1.4962212-02-09010 12-02-09019 1139 -9.9944612-02-09019 12-06-09058 641 -3.2490312-06-09058 12-06-09113 116 -0.4650212-06-09058 12-06-09113 99 -0.4653412-06-09113 12-02-09016 602 -9.4824012-02-09016 12-02-09008 873 12.6861912-02-09008 12-02-09006 943 1.0808712-02-09006 12-02-09033 822 3.4079112-02-09033 12-02-09066 917 -4.4460712-02-09066 K -06-09202 867 13.90090K -06-09202 K -06-09166 1024 -20.86354K -06-09166 K -06-09062 204 4.93669K -06-09062 K -06-09138 983 -10.55822K -06-09138 K -06-09064 442 -1.46610K -06-09064 K -06-09072 733 2.22171K -06-09072 G.I.1607 238 -5.87435Kärnan: foundation wall - tower chamber032*1*3124 032*2*3109 40 6.98720

The National Survey and CadastreRentemestervej 8DK-2400 Copenhagen NV Denmark

+45 7254 5000www.kms.dk kms@kms.dk